A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in Climate Change

Abstract

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state-reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model’s performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

Appendices

Appendix

A Normalizing the Bellman Equation

The general Bellman equation expressed in terms of the original value function $V$ is

$$\begin{aligned}&V(K_t,M_t,T_t,\varvec{\Phi }_t,t) = \, \max _{C_t,\mu _t} L_t \frac{\left( {C_t}{\big /}{L_t} \right) ^{1-\eta }}{1-\eta } {\Delta t}\ + \frac{\exp [-\delta _u {\Delta t}]}{1-\eta }\\&\quad \times \left( \text{ E }\ \left[ (1-\eta ) V \! (K_{{t + \Delta t}},M_{{t + \Delta t}},T_{{t + \Delta t}},\varvec{\Phi }_{t + \Delta t},t\!+\!\Delta t) \right] ^{\frac{1-{{\mathrm{RRA}}}}{1-\eta }} \right) ^{\frac{1-\eta }{1-{{\mathrm{RRA}}}}}. \end{aligned}$$

Using effective labor units, i.e., the definitions $c_t=\frac{C_t}{A_t L_t}$ and $k_t=\frac{K_t}{A_t L_t}$, we bring this equation to the form

$$\begin{aligned}&V(k_t A_t L_t,M_t,T_{T},\varvec{\Phi }_t,t) = \max _{c_t,\mu _t} \frac{c_{t}^{1-\eta }}{1-\eta } A_t^{1-\eta } L_t {\Delta t}+ \frac{\exp [-\delta _u {\Delta t}]}{1-\eta } \exp \Bigg [(1-\eta )g_{A,t}{\Delta t}\Bigg ]\\&\quad \times A_{t}^{1-\eta } \exp [g_{L,t}{\Delta t}] L_{t} \ \left( \text{ E }\left[ (1-\eta ) \frac{V(K_{{t + \Delta t}},M_{{t + \Delta t}},T_{{t + \Delta t}},\varvec{\Phi }_{t + \Delta t},{t + \Delta t})}{A_{{t + \Delta t}}^{1-\eta } L_{{t + \Delta t}}} \right] ^{\frac{1-{{\mathrm{RRA}}}}{1-\eta }} \right) ^{\frac{1-\eta }{1-{{\mathrm{RRA}}}}} \! . \end{aligned}$$

Table 1 Parameters of the model (base case calibration)

Fig. 5

The calibration of the rate governing CO$_2$ removal from the atmosphere. We calibrate the initial rate $\delta _{M,0}$ and the asymptotic rate $\delta _{M,\infty }$. The first line in the legend displays the parameter values chosen in our calibration. The other lines show the value of the parameter that was changed with respect to our chosen calibration

Fig. 6

The calibration of the rate governing CO$_2$ removal from the atmosphere. We calibrate the parameter $\delta ^{*}_{M}$ governing the speed of convergence from the initial to the asymptotic rate of CO$_2$ removal

The second line inserted $A_{t+1}^{1-\eta } L_{t+1}$ in the denominator of the inner bracket, which cancels with the term $A_{t+1}^{1-\eta } L_{t+1}=\exp [(1-\eta )g_{A,t}{\Delta t}] A_{t}^{1-\eta } \exp [g_{L,t}{\Delta t}] L_{t}$ inserted outside of the brackets. We divide both sides of the equation by $A_t^{{1-\eta }} L_t$ and define $V^*(k_\tau ,M_\tau ,T_\tau ,\varvec{\Phi }_\tau ,\tau )=\frac{\left. V(K_t,M_t,T_t,\varvec{\Phi }_t,t) \right| _{K_t=k_t A_t L_t}}{ A_t^{{1-\eta }}L_t}$ yielding

$$\begin{aligned} V^*(k_t,M_t,T_t,\varvec{\Phi }_t,t)&= \max _{c_t,\mu _t} \frac{c_{t}^{1-\eta }}{{1-\eta }} {\Delta t}+ \frac{\exp [(-\delta _u+g_{A,t} (1-\eta ) + g_{L,t}){\Delta t}]}{{1-\eta }}\\&\quad \times \left( \text{ E }\left[ (1-\eta ) V^*(k_{t+\Delta t},M_{ t+\Delta t},T_{ t+\Delta t}, \varvec{\Phi }_t,t+\Delta t) \right] ^{\frac{{1-{{\mathrm{RRA}}}}}{{1-\eta }}} \right) ^{\frac{{1-\eta }}{{1-{{\mathrm{RRA}}}}}}, \end{aligned}$$

and, thus, Eq. (26) and its special case equation (23).

Fig. 7

The calibration of the warming delay parameter $\sigma _{forc}$ and the parameter $\sigma _{ocean}$ connecting atmospheric and oceanic temperatures. The first line in the legend displays the parameter values chosen in our calibration. The other lines show the value of the parameter that was changed with respect to our chosen calibration

Fig. 8

Compares our simple min-quadratic approximation of the temperature difference between the atmosphere and the oceans given in Eq. (17) to the actual difference resulting from the DICE-2007 model. The noticeable difference emerging two centuries into the future causes also the slightly more pronounced difference between our and DICE’s atmospheric temperature observed after the year 200 in the earlier calibration plots. Section 5.3 discusses a more sophisticated interpolation using a Chebychev basis and a larger set of interpolation paths (see Fig. 9)

B Calibration

The section summarizes the numerical part of the calibration discussed in Sect. 3.2 whose result we presented in Sect. 5.1. Table 1 summarizes the model parameters. We calibrate our model so that the optimal time paths of CO$_2$ concentration, temperature, abatement rate, and optimal carbon tax closely resemble those predicted by the DICE-2007 model.²⁴ Figures 5 and 6 show our calibration of Eq. (16), approximating the carbon cycle. Our carbon removal rate decreases from the initial value $\delta _{M,0}$ to the asymptotic value $\delta _{M,\infty }$, where the rate of decline is characterized by $\delta ^*_M$. Figure 7 shows the calibration of the heat capacity and feedback related delay parameter $\sigma _{forc}$, and of the parameter $\sigma _{ocean}$ capturing ocean temperature related feedbacks (Eq. 13). Any individual parameter change improving the fit in one dimension worsens the fit in at least one of the other variables. Finally, Fig. 8 depicts the difference between oceanic and atmospheric temperatures in DICE-2007, and compares it to our simple quadratic approximation stated in Eq. (17).

Fig. 9

Compares our more sophisticated interpolation of the atmosphere-ocean temperature difference and the rate of carbon removal (Sect. 3.3). The circles show the decadal values resulting from the original DICE paths (data). The dashed blue line shows the (average) time trend fitted by a 30 node cubic spline. The solid lines recover the original data using the time trend and a linear, period-dependent correction based on the atmospheric carbon content

We solve the Bellman equation (23) by help of the function iteration algorithm described in Sect. 4.2. We approximate the value function $V^*$ by Chebyshev polynomials (Fig. 9). We update the basis coefficients by collocation at the Chebychev nodes spelled out in Table 2 (rectangular grid). To arrive at this final node grid, we sequentially increased the number of nodes in each dimension, until a further increase in the number of basis function no longer affected the solution. Figure 2 shows that a further increase of the node number beyond our $18\times 6\times 10\times 6=6480$ nodes has no observable effect on increasing the accuracy of our simulation. Our convergence criterion was a coefficient change of less than $10^{-4}$. The corresponding maximal relative change in the value function was less than $10^{-10}$. Figure 10 shows that a further reduction of the convergence tolerance by an order of magnitude had no effect on the optimal time paths of the variables of interest.

Table 2 Location of Collocation Nodes

Fig. 10

Robustness of the results to a decrease in the convergence tolerance

Fig. 11

Compares our simple calibration and those of the DICE 2007 and DICE 2013 to the MAGICC model. The scenarios are the same as in Fig. 3, but here we directly compare the temperature response to the radiative forcing generated in the different scenarios by all greenhouse gases. The vertical axes measure the temperature increase over pre-industrial levels in degree Celsius, starting with the same initial conditions in 2005

Fig. 12

Compares our simple calibration and those of the DICE 2007 and DICE 2013 to the MAGICC model. We show the temperature response to the remaining emission scenarios used in the 4th (SRES) and 5th (RCP) Assessment Reports of the IPCC. On the left we show the direct emission response, on the right the temperature response of the implied radiative forcing by all greenhouse gases. The vertical axes measure the temperature increase over pre-industrial levels in degree Celsius, starting with the same initial conditions in 2005

C Further Comparisons to the MAGICC Model

The present part of the appendix extends the comparison of our base case calibration and of the DICE 2007 and DICE 2013 models to the average of the AOGCM models used in the IPCC as emulated by MAGICC 6.0. We explained in Sect. 5.2 that MAGICC and the SRES and RCP scenarios model a large number of greenhouse gas emissions explicitly, and that the DICE (and our) integrated assessment model only endogenize fossil fuel based CO$_2$ emissions. Our comparison in Sect. 5.2 asks how well the integrated assessment models perform in approximating more comprehensive climate policy scenarios. Part of the failure in reproducing the temperatures of the SRES and RCP scenarios is that these simplified integrated assessment models have a fixed exogenous forcing proxy for all non-CO$_2$ greenhouse gas emissions. A more favorable comparison between the integrated assessment models and MAGICC feeds the joint radiative forcing of all greenhouse gases of the SRES and RCP scenarios to DICE and our model. Figure 11 presents the results for the same scenarios that Fig. 3 compares based on their industrial CO$_2$ emissions component. As to be expected, all models perform significantly better in reproducing the temperature response of MAGICC. In the majority of cases, our simplified model performs slightly better than the original DICE model despite our simplifications of the temperature dynamics.

Figure 12 fills in the same comparison undertaken in Figs.  3 and 11 for the remaining SRES and RCP scenarios. The SRES A1F is a more fossil fuel intensive scenario and, like RCP 8.5 in the new scenarios, it is not a candidate for an optimal policy as it significantly exceeds 4$^\circ$C before the end of the century. RCP 3 is the other extreme. Here CO$_2$ emissions become negative already in the current century as a consequences of major immediate reductions in combination with carbon capture. Again, it is not a candidate for an optimal policy—without significantly altering abatement cost and damage assumptions and introducing cheap carbon sequestration. The major divergence between our base case calibration and the MAGICC temperature in the emissions based graph on the left hand side is a consequence of the major reduction of non-fossil fuel based emission in MAGICC not accounted for in the integrated assessment models: the graph on the right hand side takes the radiative forcing of all emissions into account and delivers a temperature response closely resembling that of MAGICC.